Geometry and Combinatorics

Speaker: Prajwal Udanshive (Western)

"Katz's proof of Rota-Heron-Welsh part III"

Time: 15:30

Room: MC 108

part 3 of the learning seminar

Colloquium

Speaker: Rasul Shafikov (Western)

"Open problem: nonexistence of Levi-flats in CP^2"

Time: 15:30

Room: MC 107

An intriguing open problem in complex geometry is to construct an example or prove nonexistence of a real analytic (or smooth), closed (i.e, compact without boundary), Levi-flat hypersurface in the complex projective plane $CP^2$. This question appeared in the context of foliation theory as a problem of the existence of nontrivial minimal sets. Nonexistence of Levi-flats in $CP^n$ for$ n > 2$ was proven by several authors, but the problem remains open for $n=2$. I will give the necessary background concerning Levi-flat hypersurfaces and outline three (mostly) self-contained proofs of the nonexistence results in $CP^n$ , $n > 2$. Two of the three proofs are due to Lins Neto, the third one is due to Siu. These proofs are genuinely different, and it is remarkable that all of them fail in dimension 2 for different reasons.

Graduate Seminar

Speaker: Saleh Ahmed (Western)

"What is Rigid Geometry and Why Should We Care?"

Time: 16:30

Room: MC 108

In short, rigid analytic geometry is a form of analytic geometry over nonarchimedean fields where the spaces of study are constructed by gluing polydiscs. In this talk, we will give an overview of this construction, explain how rigid geometry serves as an analogue to complex analytic geometry, and why attempting to naively mimic complex analytic geometry fails. We will conclude with a discussion on some applications.

Geometry and Combinatorics

Speaker: (Western)

"Family day -- no seminar"

Time: 14:30

Room: MC 108

Geometry and Combinatorics

Speaker: Mieke Fink (Western/Osnabrück)

"Bergman fans of cosmological hyperplane arrangements"

Time: 15:30

Room: MC 108

Cho-Matsumoto and Kita-Yoshida introduced intersection forms on the twisted (co)homology of hyperplane arrangement complements. Recently, Thomas Lam gave a closed form of the pairing in terms of the matroid of the arrangement. We compute the Bergman fan of some arrangements of interest in cosmology in order to apply Lam's formula in this setting.

Graduate Seminar

Speaker: Elias Vandenberg (Western)

"A beginner's guide to tangent categories"

Time: 16:30

Room: MC 108

Tangent categories are special categories that generalize the structure of the category of smooth manifolds. Originally defined for smooth manifolds by Rosický in 1984, tangent categories were rediscovered and generalized to arbitrary categories by Cockett and Cruttwell in 2014. In the twelve years following this paper's publication, the theory of tangent categories has undergone rapid development, and tangent categories have been studied in a wide range of settings, including (synthetic) differential geometry, algebraic geometry, differential linear logic, and machine learning.

We will give a concrete explanation of how the theory of tangent categories can be developed from scratch using basic tools from differential geometry (manifolds and tangent bundles) and category theory (categories, functors, and natural transformations). We'll also provide some examples to illustrate the relationship between tangent categories and other mathematical disciplines like differential/algebraic geometry.